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# Commensurate 4*a*_{0}-period charge density modulations throughout the Bi_{2}Sr_{2}CaCu_{2}O_{8+x} pseudogap regime

Contributed by J. C. Séamus Davis, September 1, 2016 (sent for review July 13, 2016; reviewed by Marc-Henri Julien and J. Zaanen)

## Significance

Strong Coulomb interactions between electrons on adjacent Cu atoms result in charge localization in the cuprate Mott-insulator state. When a few percent of electrons are removed, both high-temperature superconductivity and exotic charge density modulations appear. Identifying the correct fundamental theory for superconductivity requires confidence on whether a particle-like or a wave-like concept of electrons describes this physics. To address this issue, here we take the approach of using the phase of charge modulations, available only from atomic-scale imaging. It reveals a universal periodicity of the charge modulations of four crystal unit cells. These results indicate that the particle-like concept of strong interactions in real-space provides the intrinsic organizational principle for cuprate charge modulations, implying the equivalent for the superconductivity.

## Abstract

Theories based upon strong real space (*r*-space) electron–electron interactions have long predicted that unidirectional charge density modulations (CDMs) with four-unit-cell (4*a*_{0}) periodicity should occur in the hole-doped cuprate Mott insulator (MI). Experimentally, however, increasing the hole density *p* is reported to cause the conventionally defined wavevector *Q*_{A} of the CDM to evolve continuously as if driven primarily by momentum-space (*k*-space) effects. Here we introduce phase-resolved electronic structure visualization for determination of the cuprate CDM wavevector. Remarkably, this technique reveals a virtually doping-independent locking of the local CDM wavevector at _{2}Sr_{2}CaCu_{2}O_{8}. These observations have significant fundamental consequences because they are orthogonal to a *k*-space (Fermi-surface)–based picture of the cuprate CDMs but are consistent with strong-coupling *r*-space–based theories. Our findings imply that it is the latter that provides the intrinsic organizational principle for the cuprate CDM state.

Strong Coulomb interactions between electrons on adjacent Cu sites result in complete charge localization in the cuprate Mott insulator (MI) state (1). When holes are introduced, theories based upon the same strong ** r**-space interactions have long predicted a state of unidirectional modulation of spin and charge (2⇓⇓⇓⇓⇓⇓⇓⇓–11), with lattice-commensurate periodicity for the charge component. Experimentally, it is known that even the lightest hole doping of the MI state produces nanoscale clusters of charge density modulations (CDMs) (12, 13), which implies immediately that

**space interactions predominate. However, with increasing hole density**

*r*-*p*, the conventionally defined wavevector

*Q*_{A}of the CDM is reported to increase (14) or diminish (15) continuously as if driven primarily by

**-space (Fermi surface) effects. Distinguishing between the**

*k***-space and**

*r***-space theoretical perspectives is critical to identifying the correct fundamental theories for the phase diagram and Cooper pairing mechanism in underdoped cuprates. Here we introduce an approach to this challenge by applying phase-resolved electronic structure visualization (16⇓–18) in combination with the technique of phase demodulation–residue minimization, to explore the CDM wavevector. Using these methods, we visualize the phase discommensurations (19) and their influence on the doping dependence of both the conventionally defined**

*k*

*Q*_{A}and the fundamental local wavevector

*Q*_{0}of the underdoped cuprate CDM state.

## CDMs in the Pseudogap Phase

As holes are introduced into the CuO_{2} plane of the MI, the first nonmagnetic state to appear is the pseudogap (PG) phase (Fig. 1*A*). It contains nanoscale CDM clusters (12, 13) even at lowest hole-density *p*; near *p* = 0.06 these CDM clusters percolate and the superconducting state appears (12). X-ray scattering experiments (15) now report a robust CDM state throughout the range 0.07 < *p* < 0.17 spanning the pseudogap regime. Both the PG and the CDM states terminate somewhere near *p* ∼ 0.19 and give way to a simple d-wave superconductor. A fundamental reason for the great difficulty in understanding this complex phase diagram has been the inability to discern the correct theoretical starting point. Should one focus on the intense ** r**-space electron–electron interactions that form the basis for the parent MI state? Or should one focus upon a Fermi surface of momentum space eigenstates representing delocalized electrons?

A new opportunity to address these questions has emerged recently, through studies of the CDM phenomena now widely observed in underdoped cuprates (15, 20, 21). Pioneering studies of CDMs in La_{2-x}Ba_{x}CuO_{4} and La_{2-x-y}Nd_{y}Sr_{x}CuO_{4} near *p* = 0.125 discovered charge modulations of period 4*a*_{0} or ** Q** = ((1/4,0);(0,1/4))2

*π*/

*a*

_{0}(14, 21). The initial intuitive explanation for a periodicity that was half that predicted from a Hartree–Fock momentum space treatment invoked an

**-space model involving local magnetic moments whose antiferromagnetic order becomes frustrated upon hole doping. A variety of powerful theoretical techniques (2⇓⇓⇓⇓⇓⇓⇓⇓–11) support this strongly interacting**

*r***-space viewpoint. In the interim, however, CDM phenomena have been discovered within the pseudogap regime of many other underdoped cuprates (15, 20, 21). In these studies, the magnitude of the conventionally determined CDM wavevector**

*r***is reported to increase/diminish with increasing**

*Q**p*, as if evolution of momentum space electronic structure with carrier density is the cause. Thus, distinguishing between an

**-space–based and a Fermi-surface–based theoretical approach to the cuprate CDM remains an outstanding and fundamental challenge and one that is key to the larger issue of controlling the balance between different electronic phases. The reason is that, in the**

*r***-space context (22⇓⇓–25), competition for spectral weight at the Fermi surface between different electronic states including the superconductivity is a zero sum game: Suppressing one state amplifies another and vice versa. By contrast, in the strong interaction**

*k***-space context (2⇓⇓⇓⇓⇓⇓⇓⇓–11), the physics of holes doped into an antiferromagnetic MI yield “intertwined” states (4, 8⇓⇓–11, 26, 27) including superconductivity, where closely related ordered states are generated simultaneously by the same microscopic interactions.**

*r*## CDMs and Phase Discommensurations

Understanding the cuprate CDM phenomenology has proved challenging (28⇓–30) because its ** q**-space peaks are typically broad with spectral weight distributed over many wavevectors (15⇓⇓⇓⇓⇓–21) and also because of the form factor symmetry (17, 31⇓–33). For example, Fig. 1

*B*and

*C*shows a typical image of the electronic structure of underdoped Bi

_{2}Sr

_{2}CaCu

_{2}O

_{8}

_{+x}

**for bias voltage**

*r**V*. The CDM exhibits a

*d*-symmetry form factor meaning that modulations at the

*x*-axis–oriented planar oxygen sites

*O*

_{x}(

**) are**

*r**π*out of phase with those at

*y*-axis–oriented oxygen sites

*O*

_{y}(

**), as shown schematically by the overlay in Fig. 1**

*r**B*. Thus, the complex Fourier transforms

*O*

_{x}(

**);**

*r**O*

_{y}(

**) that are derived (17, 33) from**

*r**d*-symmetry form factor Fourier amplitudes

*D*(

*SI Text*,

*d-Symmetry Form Factor of CDMs*). One sees directly the wide range of

**values that exist under each CDM peak in**

*q**D*). Such broad peaks indicate quenched disorder of the CDM but with two quite distinct possibilities for the identity of the fundamental ordered state: (

*i*) an incommensurate CDM state whose wavevector

**evolves continuously along with the Fermi wavevector**

*Q*

*k*_{F}(e.g., Fig. 1

*E*) but is perturbed by disorder or (

*ii*) a commensurate CDM with constant fundamental wavevector

*Q*_{0}driven by strong-coupling

**-space effects (e.g., Fig. 1**

*r**F*), but whose wavevector defined at the maximum of the fitted Fourier amplitudes,

McMillan (19) defined a “discommensuration” as a defect in a commensurate CDM state where the phase of the CDM jumps between discrete lattice-locked values. For example, consider a sinusoidal modulation in one spatial dimension with a commensurate period 4*a*_{0}:*A* is the amplitude, and *A*, *i* and *ii*. When a commensurate CDM (Eq. **1**) is frustrated by Fermi-surface–based tendencies, a regular DC array allows more/fewer modulations to be accommodated through successive jumps in phase (Fig. 2 *A*, *ii* and ref. 19). The result is a new phase-averaged wavevector *A*, *ii*. In this case, the difference in slope between the commensurate and phase-averaged wavevectors, *A*, *iii*) of such a nominally incommensurate phase. Note that such a DC array does not affect the correlation length of the CDM even though it does shift the Fourier amplitude-defined wavevector *A*, *iii*). In contrast, when the combined phase jumps of all of the DCs average to zero (Fig. 2 *B*, *i* and *ii*), the phase-averaged wavevector *B*, *iii*). However, in the most realistic case where disorder in the CDM amplitude and the random spatial arrangement of DCs coexist, *C*, *i* and *ii*; *SI Text*, *Heterogeneity and Demodulation Residue*; and *SI Text*, *Statistical Analysis: Two-Dimensional Fitting*).

How then can one correctly determine the fundamental *A*, *iii* may be detectable through the observation of satellite peaks at *SI Text*, *Heterogeneity and Demodulation Residue*; and *SI Text*, *CDM Commensurability in Underdoped BSCCO*). Phase-sensitive transmission electron microscopy can achieve DC visualization (34), whereas coherent X-ray diffraction might (35), but neither one has been used on cuprates. Instead, we consider CDM visualization using spectroscopic imaging scanning tunneling microscopy (20) because it offers full access to both the amplitude and phase of *q*, using the demodulation residue*A* and *B*. The resulting wavevector *SI Text*, *Heterogeneity and Demodulation Residue*), and determination of this

## SI Text

*d*-Symmetry Form Factor of CDMs.

For a given BSCCO sample spectroscopic imaging scanning tunneling microscopy (SI-STM) measurement of the *Z* map, where *Z*(** r**,

*E*) =

*g*(

**, +**

*r**E*)/

*g*(

**, −**

*r**E*), we calculate the dFF

*Z*-map Fourier transform

*Z*-map data at oxygen sites. We choose

### Heterogeneity and Demodulation Residue.

The CDMs observed in cuprates are short ranged, and disorder in the modulations challenges the measurement of their orientation, symmetry, and wavevector. In Fig. S1 we demonstrate the effect of disorder on two CDM scenarios from Fig. 2. To generate DCs themselves in our simulations, e.g., in Fig. 2, we rely on a Ginzburg–Landau description of CDMs detailed in *SI Text*, *CDM Commensurability in Underdoped BSCCO*.

In the case of the DC lattice (Fig. S1*A*) we consider variation in sizes of commensurate domains, keeping the incommensurability fixed (Fig. S1*B*), and we add smooth variation of CDM amplitude (Fig. S1*A*). The initially singular Fourier amplitude satellite peaks (Fig. 2 *A*, *iii*) can completely disappear in the generated noise (Fig. S1*C*). Observation of the DC lattice by amplitude-measuring scattering probes can therefore be hindered by disorder, as is demonstrably the case in layered dichalcogenides. In the case of randomly positioned DCs having random phase slips (Fig. 2 *B*, *i* and Fig. S1*E*), we add disorder in amplitude, using smooth variation (Fig. S1*D*). The resulting Fourier amplitudes (Fig. S1*F*) show a broad irregular peak with high values at multiple wavevectors. This situation is characteristic of BSCCO SI-STM data and can be generated in the case of a periodic DC array (Fig. S1*B*), e.g., by adding random DC with random phase slips. To avoid ambiguities in determining the average wavevector from Fourier amplitudes (below and *SI Text*, *Statistical Analysis: Two-Dimensional Fitting*), we introduce a phase-sensitive measurement based on demodulation residue, which robustly determines the average wavevector with high precision (Fig. 2 *C*, *ii* and Fig. S1*F*).

To describe the phase-sensitive measurement of the average wavevector, we consider a generally short-ranged 1D smooth modulation *B*, *ii* is an example representing BSCCO; however, the following analysis is relevant for any disordered *x* with zero average value.

Generally, any measurement samples the modulation in some finite region *L* for a typical Gaussian disorder.

According to Eq. **1**, we can interpret *x* term that is proportional to *Q* therefore has the minimal amount of spatial modulation. In an example of uniform modulation (long-range CDW with no disorder), the optimally demodulated function is strictly a constant equal to the amplitude of the modulation, and

In the presence of disorder, phase-optimized *Q* rigorously equals *i*) when spatial fluctuations of complex amplitude *ii*) when the complex phase spatially fluctuates on shorter scales than the complex amplitude does, even if the amplitude fluctuates strongly. The CDMs observed in SI-STM images of BSCCO have primarily phase disorder, putting them close to the first limit.

We emphasize that even if the randomly disordered modulation strongly violates conditions for both limits *i* and *ii*, the error *q* grows the complex phase of demodulation varies faster in space (due to the *ii* is satisfied, making *S* and *R* can deviate from each other only within a region of width *R* (Fig. 2 and Fig. S1).

To determine *SI Text*, *Statistical Analysis: Two-Dimensional Fitting*). The use of phase-optimized *Q* is especially necessary when the amplitude peak is asymmetric, in which case *C*, *ii*. With a roughly symmetric peak the *Q* and give a good estimate of *F*. One way to generate the former case with large mismatch of *G*, used to obtain Fig. 2 *C*, *ii*, compared with the case in Fig. S1*E*, used to obtain Fig. S1*F*. In BSCCO SI-STM data, the mismatch of commensurate *Q* and doping-dependent (33) *H*.

Additionally, in two dimensions there are topological point-like defects, dislocations (Fig. S2*A*), where the considered demodulation *SI Text*, *Demodulation in Two Dimensions, Smoothing, and Optimization Error*) is therefore smooth at the position of dislocation, even though it contains derivatives. Crucially, the local disturbance in *B* and *C*). The dislocations therefore generally do not influence the demodulation measurement of the phase-optimized wavevector.

### Statistical Analysis: Two-Dimensional Fitting.

Given the dFF data, we focus on a single characteristic intensity peak (out of two, *x* and the *y* axis, respectively) by restricting *B*). These restricted data are simply referred to as “dFF data” and labeled

The Fourier amplitude of the dFF, *s* in orthogonal directions. For example, the data presented in Fig. S3*A* are best fitted by the shifted Lorentzian centered at

We, however, observe that the fit is quite poor, with values of adjusted *R*^{2} of about 0.7, which corroborates visual inspection, e.g., in Fig. 3*C*. The culprit for the poor fit is data having multiple local (single-pixel) peaks in an asymmetric distribution. We confirm this by showing that statistically these peaks cannot be described by the simple smooth single-peaked distribution.

A standard tool for analyzing the nature of the fit is the normal quantile–quantile (QQ) plot, shown for all of the samples in Fig. S4. The fit residuals, i.e., the set of differences between a value

This deviation is statistically significant, according to the comparison of the highest value of the fit residual with the probability distribution for the highest value drawn from a normal distribution, this normal distribution best describing the fit residuals. The observed highest residual value typically lies within 3–6 SDs away from the center of this probability distribution.

### Demodulation in Two Dimensions, Smoothing, and Optimization Error.

To apply our analysis to 2D data, the SI-STM *Z*-map dFF *A*) (48). The analysis of CDM requires smoothing because one needs to include only the data within a single broad intensity distribution (the one around either the *x* or the *y* axis) from the Fourier data of *A*). Note that the demodulated pattern *B*).

The demodulation residue in two dimensions is a vector with components *C* and *D*) and in field-of-view of finite size *L* is a direct generalization of the one-dimensional case:*X* and *Y* intensity peaks in Fourier data of

As discussed in *SI Text*, *Heterogeneity and Demodulation Residue*, the phase-optimized wavevector gives a linear fit *B* and *E* for 1D examples). To estimate the error of the obtained slope *A*, *ii* and *B*, *ii* the *F*) by simple conversion of units from

### CDM Commensurability in Underdoped BSCCO.

When we extract the phase-optimized wavevector, representing the average wavevector of CDMs, we find values consistent with commensurate *SI Text*, *Summary of Results for All Samples*). We now demonstrate the consistency of this finding and simultaneously show that the local wavevector is also *A* (also Fig. 3*F* and Fig. S1*H*), shows multiple domains with CDM phase close to the commensurately pinned values *B*), areas where CDM order parameter amplitude is high consistently show commensurate patches of a typical *B* and *F*) and reveal these patches forming commensurate domains. Every patch individually is therefore aligned with an underlying lattice, an effect observed already in ref. 49, but the patches are shifted in position with respect to each other by values *a*_{0}, 2*a*_{0}, or 3*a*_{0}.

As in our earlier analysis (18) we here observe that CDM order parameter phase fluctuations are dominated by phase vortices (white circle in Fig. S6*A*), i.e., CDM dislocations, which seems to indicate an incommensurate CDM. In commensurate CDMs, dislocations also exist as points where multiple DCs meet and provide a total of *SI Text*, *Demodulation in Two Dimensions, Smoothing, and Optimization Error*) approaches the commensurate domain size, the DC meeting point smooths into a phase vortex in Fig. S6*A*.

Although our data show local wavevector *F* and Figs. S1*H* and S6), in principle there could be an underlying DC lattice and an incommensurate average wavevector observed over distances larger than the SI-STM field-of-view. The error in phase-optimized wavevector *SI Text*, *Demodulation in Two Dimensions, Smoothing, and Optimization Error*. We note that by definition of the phase-optimized wavevector, an ideal DC lattice would appear as a seesaw in phase of the CDM order parameter; e.g., see deviation of phase argument from best-fit line in Fig. 2 *A*, *ii*. We do not see such features in SI-STM images.

To generate DCs in our simulations (Fig. 2 and Figs. S1 and S8), we use a Ginzburg–Landau order parameter theory for CDMs, which in cuprates has square lattice symmetry (50). The theory in ref. 50 deals with the commensurate CDM state in a region around doping 1/8, but does not deal with DCs. Compared with well-known CDW systems in layered dichalcogenides (51, 52), the energy term describing the local commensuration tendency appears as a quartic term instead of as a cubic. We focus on the CDM phase mode and find an ideal DC lattice as a possible ground state (19). The quartic term prefers *n* integer, and separation distance *B* shows an example of

### Summary of Results for All Samples.

Fig. S7 shows the demodulation residue for modulations along the *x* axis, calculated for all analyzed samples (for *y*-axis modulations, see Fig. 4). For underdoped samples a single phase-optimized wavevector is apparent. For higher dopings the distinction is not so clear and the optimum is within a larger area of Fourier space, in accord with the weakening of the dFF component at those dopings. The results are robust to changes of filtering cutoff *SI Text*, *Demodulation in Two Dimensions, Smoothing, and Optimization Error*).

### Global CDM Lattice Commensurability.

We introduce the following hypothesis: Differences in measured CDM wave vectors over different cuprate families are due to different arrangements of DCs in the underlying *A*, *iii* and Fig. S1*C*), and this effect is already recognized in layered dichalcogenides (34).

In BSCCO, our findings of *SI Text*, *CDM Commensurability in Underdoped BSCCO*) are simulated in a 1D cut by a random distribution of commensurate domains of typical size around 8*a*_{0}, with DCs whose phase slips are random multiples of *D*, *E*, and *G* and Fig. 2 *B*, *i* and *ii*). The irregular Fourier amplitude peak with phase-optimized wavevector *C*, *ii* and Fig. S1*F*) have similar full-width *B*).

In the case of YBa_{2}Cu_{3}O_{7-x}, existing X-ray measurements (15) offer two restrictions: (*i*) The CDM intensity peak is centered on wavevector *ii*) the intensity peak has full-width *A*, *iii* shows how in absence of disorder a DC lattice generates a sharp Fourier peak at *C*).

Neutron scattering in La_{2-x-y}Nd_{x}Sr_{y}CuO_{4} (ref. 39 and references therein) shows

X-ray scattering on La_{2-x}Ba_{x}CuO_{4} (14) finds a peak near _{2-x-y}Nd_{x}Sr_{y}CuO_{4} scenario can simply be caused by any of several factors such as increased CDM amplitude variation, more randomness in positions of DCs within their lattice, or additional random DCs that average to zero phase slip and occur on top of the underlying DC lattice.

We uncovered the problem of irregular Fourier amplitude distribution within a broad CDM intensity peak and it is presently unclear whether this property can be observed in other cuprates. In the Bismuth-based cuprate family resonant X-ray scattering experiments (43, 44) have a high wavevector resolution and clearly show these broad irregular Fourier amplitude distributions. On the other hand, X-ray experiments on YBCO show smoother Fourier intensity peaks, but one must focus on the wavevector resolution: Typically (45, 53) there are only about 5 distinct amplitude values in a 1D cut of *q* space through the intensity peak, which washes out the possible irregularity. In comparison, SI-STM data used here have of order 50 amplitude values in a 2D profile of the intensity peak (Fig. 3*D*).

## Phase-Resolved Imaging and Phase Demodulation Residue Analysis

We apply demodulation residue analysis to study 2D short-range ordered CDM images typical of underdoped Bi_{2}Sr_{2}CaCu_{2}O_{8+x}, e.g., Fig. 3*A* at *p* = 0.06. Fig. 3*B* shows the *d*-symmetry form factor Fourier amplitude *A*. The amplitude and phase of modulations in *x*, *y*. We define the demodulation residue *SI Text*, *Demodulation in Two Dimensions, Smoothing, and Optimization Error*) over a wide range in the Fourier space inside the dashed box in Fig. 3*B* and use it as a phase-sensitive metric for deciding how close each *B* and figure 3.7b of ref. 20) and not well fitted by a smooth peak (Fig. 3*C* and *SI Text*, *Statistical Analysis: Two-Dimensional Fitting*). Hence we will seek the minimum of *A* and *B* we calculate the *E* (*SI Text*, *Demodulation in Two Dimensions, Smoothing, and Optimization Error*). In Fig. 3*E* we plot the value of both *d*-symmetry form factor (dFF) Fourier transform *D*. This shows that the procedure singles out one wavevector for the CDM in the *y* direction with a nearly vanishing demodulation residue, a gap between this **-**minimizing *D*. Instructively, this demodulation residue-minimizing **-**minimization approach is that it singles out the phase-averaged *SI Text*, *Heterogeneity and Demodulation Residue*; and *SI Text*, *Statistical Analysis: Two-Dimensional Fitting*). Most remarkably, we find that ** r** space where the CDM phase is locked to the four expected discrete values [

*F*], forming locally commensurate

*SI Text*,

*CDM Commensurability in Underdoped BSCCO*). In this highly typical case of a Bi

_{2}Sr

_{2}CaCu

_{2}O

_{8+x}

*B*, confirming the fundamental

Given this demonstrated capability of ** Q** from short-range CDM data, we next turn our attention to the doping dependence of fundamental

_{2}Sr

_{2}CaCu

_{2}O

_{8+x}(

*SI Text*,

*Summary of Results for All Samples*). Fig. 4

*A*contains two side-by-side panels; Fig. 4

*A*,

*Left*shows measured

*A*,

*Right*is the measured

*y*-axis modulations. Fig. 4

*A*–

*E*then shows a series of such pairs of measured

_{2}Sr

_{2}CaCu

_{2}O

_{8+x}hole-densities

*p*= 0.06, 0.08, 0.10, 0.14, 0.17. In all cases, the demodulation residue-minimizing process clearly singles out the minimized values in

*π*/

*a*

_{0}point (marked by a cross in Fig. 4

*A*–

*E*,

*Right*,

*F*is that the measured magnitudes of the average wavevectors

_{2}Sr

_{2}CaCu

_{2}O

_{8+x}CDM are all indistinguishable from the lattice commensurate values

*SI Text*,

*Summary of Results for All Samples*). Moreover, the largest deviation of the conventional amplitude-derived

*SI Text*,

*Heterogeneity and Demodulation Residue*).

## Ubiquity of Lattice-Commensurate CDMs

Comparison of this result with reports of a preference for a CDM periodicity of 4*a*_{0} in YBa_{2}Cu_{3}O_{7-x}-based heterostructures (36), in the NMR-derived view of the lattice-commensurate CDM in YBa_{2}Cu_{3}O_{7-x} (37), and in the pair density wave state of Bi_{2}Sr_{2}CaCu_{2}O_{8+x} (38) points to growing evidence for a unified phenomenology of lattice-commensurate CDMs across disparate cuprate families. Of course, the wavevectors _{2}Cu_{3}O_{7-x} and La_{2}Sr(Ba)CuO_{4} families evolve continuously with doping and appear generally incommensurate (14, 15). However, DC configurations of the type in Fig. 2*A* will result in such an incommensurate average wavevector *SI Text*, *Global CDM Lattice Commensurability*); a related hypothesis has long been considered (39). Our application of the classic theory of CDM DC disorder (19) (Fig. 2) with CDM phase-resolved imaging and analysis reveals this as the correct picture for Bi_{2}Sr_{2}CaCu_{2}O_{8+x}. This finding motivates the hypothesis that doping dependence of

The **-**minimization technique introduced here is designed to use the additional CDM information available only from phase-resolved imaging (16, 17, 33). Remarkably, it reveals a doping-independent locking of the phase-averaged CDM wavevector to a lattice commensurate wavevector _{2}Sr_{2}CaCu_{2}O_{8+x} (Fig. 4). Moreover, we directly detect the CDM DCs between phase-locked commensurate regions that generate this situation (Fig. 3*F*). These observations have significant fundamental consequences for understanding the mechanism of the cuprate CDM state. They are orthogonal to a weak-coupling ** k**-space–based picture for CDM phenomena, in which the fundamental wavevector should increase or decrease monotonically with doping or should evolve in discrete jumps even with “lattice locking.” Moreover, the commensurability is intractable as a perturbative effect of interactions in the

**-space picture (40). By contrast, a lattice-commensurate CDM state has been obtained comprehensively in different types of strong-coupling**

*k***-space–based theories (2⇓⇓⇓⇓⇓⇓⇓⇓–11). For underdoped Bi**

*r*_{2}Sr

_{2}CaCu

_{2}O

_{8+x}at least, our data are far more consistent with such lattice-commensurate strong-coupling

**-space theories being the intrinsic organizational principle for the cuprate CDM phenomena. Furthermore, nanoscale clusters of lattice-commensurate CDMs are the first broken-symmetry state to emerge at lightest hole doping (12, 13), multipletransport and spectroscopic measurements of cuprate quasiparticles have recently been demonstrated to be quite consistent with lattice-commensurate**

*r***-space theories (41), and YBa**

*r*_{2}Cu

_{3}O

_{7-x}NMR studies (37, 42) are also most consistent with them. Explorations of universality of lattice commensurability of CDMs in other cuprate compounds can now be pursued using these phase-resolved imaging and

**-**minimization techniques.

## Acknowledgments

A.M. acknowledges support from the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Science and Engineering, under Award DE-SC0010313; E.-A.K. acknowledges Simons Fellow in Theoretical Physics Award 392182; S.D.E. acknowledges funding from Engineering and Physical Sciences Research Council Grants EP/G03673X/1 and EP/1031014/1; M.H.H. acknowledges support from the Moore Foundation’s Emergent Phenomena in Quantum Systems Initiative Grant GBMF4544; S.-i.U. and H.E. acknowledge support from a Grant-in-Aid for Scientific Research from the Ministry of Science and Education (Japan) and the Global Centers of Excellence Program for the Japan Society for the Promotion of Science. J.C.S.D. acknowledges gratefully the hospitality and support of the Tyndall National Institute, University College Cork. Experimental studies were supported by the Center for Emergent Superconductivity, an Energy Frontier Research Center, headquartered at Brookhaven National Laboratory and funded by US Department of Energy Grant DE-2009-BNL-PM015.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: jcseamusdavis{at}gmail.com or eun-ah.kim{at}cornell.edu.

Author contributions: A.M., M.J.L., and E.-A.K. designed research; J.C.S.D. and E.-A.K. supervised the project; A.M., K.F., S.D.E., M.H.H., M.J.L., and E.-A.K. performed research; H.E. and S.-i.U. contributed new reagents/analytic tools; and A.M., K.F., J.C.S.D., M.J.L., and E.-A.K. wrote the paper.

Reviewers: M.-H.J., Laboratoire National des Champs Magnetiques Intenses–Grenoble; and J.Z., Leiden University.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1614247113/-/DCSupplemental.

Freely available online through the PNAS open access option.

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*a*

_{0}charge modulations in Bi

_{2}Sr

_{2}CaCu

_{2}O

_{8+x}

## Citation Manager Formats

*a*

_{0}charge modulations in Bi

_{2}Sr

_{2}CaCu

_{2}O

_{8+x}

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